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In other words, what do we know about any basis for R? What are its properties?

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- Thread starter jsgoodfella
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- #1

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In other words, what do we know about any basis for R? What are its properties?

- #2

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Well, we know that each matrix is pxp, because Z is p-dimensional. The dimension of this space of matrices is p-squared (for instance there are 4 basis matrices for the 2x2 case).

But how can we know that each matrix in the p^2 dimensional basis is a linear matrix?

- #4

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But how can we know that each matrix in the p^2 dimensional basis is a linear matrix?

Huh, what do you mean?? Every matrix is linear.

- #5

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Huh, what do you mean?? Every matrix is linear.

I'm thinking of the general basis for p^2 - If p is two, then [{0,1};{0,0}] could be a matrix in the basis which doesn't represent a linear transformation. Do we just assume they are all linear?

On another note, is there a linear transformation that maps every vector in a vector space to zero?

- #6

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I'm thinking of the general basis for p^2 - If p is two, then [{0,1};{0,0}] could be a matrix in the basis which doesn't represent a linear transformation. Do we just assume they are all linear?

It does represent a linear function, namely f(x,y)=(y,0). Every matrix represents a linear function, that's why matrices are so important.

On another note, is there a linear transformation that maps every vector in a vector space to zero?

Yep, that is a linear transformation. It's the analog of the zero matrix.

- #7

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Thanks for your help!

Just to make sure, the dimension is p-squared, right?

Just to make sure, the dimension is p-squared, right?

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Right!

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